The fixed point property in JB∗-triples and preduals of JBW∗-triples

We prove that for every member X in the class of real or complex JB∗-triples or preduals of JBW∗-triples, the following assertions are equivalent:(1)X has the fixed point property.(2)X has the super fixed point property.(3)X has normal structure.(4)X has uniform normal structure.(5)The Banach space of X is reflexive. As a consequence, a real or complex C∗-algebra or the predual of a real or complex W∗-algebra having the fixed point property must be finite-dimensional.